zsz_natCast_le (Generic counted loops over item arrays)

Proved

NLogN.zsz_natCast_le

by xuanji · Oct 17, 2026 · Mathlib 0df444a (Lean v4.33.1)

complexity-theoryinteger-multiplicationkappa-improvement

The following holds (Lean statement, in the scope of module NLogN.SSLoops):

theorem NLogN.zsz_natCast_le {j k : ℕ} (h : j ≤ k) : zsz (j : ℤ) ≤ zsz (k : ℤ)

Context. The lemma belongs to the part Generic counted loops over item arrays.

Formalization Note. This lemma belongs to layer 5 of the Lean formalization (integer-multiplication-kappa campaign) of the OpenAI preprint Integer multiplication below $n\log n$ : the uniform item-language machine (register programs over the NLogN multitape framework) assembling the Proposition 14 / Lemma 13 / Proposition 18 pieces, the resampling and chirp stages and the cost recurrences into one fixed machine, culminating in NLogN.KFTop.kappa_bound : IntMul.KappaBound (1 / 2 ^ 182) (the bound of Def_IntMul_MultitapeModel). All names refer to the definition bundles up to IntMul_Kappa_K_Defs33.

Preamble

-- OpenAI-preprint / κ-improvement development (integer-multiplication-kappa), generated by skeleton subtraction.

import Definitions.Def_IntMul_Kappa_K_Defs33

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open NLogN
open IntMul
variable {P : ℕ} {I : List Sym} {iprog : Fin P → IStmt NS P}

Formal statement

theorem NLogN.zsz_natCast_le {j k : ℕ} (h : j ≤ k) : zsz (j : ℤ) ≤ zsz (k : ℤ) := by sorry

Source

OpenAI, Integer multiplication below n log n (preprint, 23 Sep 2026), https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Integer-multiplication-below-n-log-n-September-23-2026. Lemma `NLogN.zsz_natCast_le` of the Lean development NLogN/SSLoops.lean (lines 32–33), integer-multiplication-kappa campaign.

View graph